scholarly journals Stability of ternary reaction-diffusion dynamical systems

2011 ◽  
pp. 245-268 ◽  
Author(s):  
Salvatore Rionero
Keyword(s):  
Dynamical Systems ◽  
Reaction Diffusion ◽  
Ternary Reaction

LDG method for reaction–diffusion dynamical systems with time delay

2011 ◽  
Vol 217 (22) ◽  
pp. 9173-9181 ◽  
Author(s):  
Dongfang Li ◽  
Chengjian Zhang ◽  
Hongyu Qin
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Reaction Diffusion ◽  
Ldg Method

OSCILLATIONS ON THE EDGE OF CHAOS VIA DISSIPATION AND DIFFUSION

2007 ◽  
Vol 17 (05) ◽  
pp. 1531-1573 ◽  
Author(s):  
MAKOTO ITOH ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Reaction Diffusion Equations ◽  
Nonlinear Dynamical ◽  
Edge Of Chaos ◽  
Secondary Purpose ◽  
Spatio Temporal ◽  
And Diffusion

The primary purpose of this paper is to show that simple dissipation can bring about oscillations in certain kinds of asymptotically stable nonlinear dynamical systems; namely when the system is locally active where the dissipation is introduced. Furthermore, if these nonlinear dynamical systems are coupled with appropriate choice of diffusion coefficients, then the coupled system can exhibit spatio-temporal oscillations. The secondary purpose of this paper is to show that spatio-temporal oscillations can occur in spatially discrete reaction diffusion equations operating on the edge of chaos, provided the array size is sufficiently large.

scholarly journals Wave pinning in strips

2006 ◽  
Vol 136 (5) ◽  
pp. 971-995 ◽  
Author(s):  
Karsten Matthies ◽  
C. E. Wayne
Keyword(s):  
Dynamical Systems ◽  
Parameter Space ◽  
Heterogeneous Media ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations

We consider the existence of stationary or pinned waves of reaction–diffusion equations in heterogeneous media. By combining averaging, homogenization and dynamical-systems techniques we prove under mild non-degeneracy conditions that if the heterogeneity is periodic with period ε, pinned solutions persist at most for intervals in parameter space whose length is O(e−c/√ε).

scholarly journals Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

2005 ◽  
Vol 135 (4) ◽  
pp. 703-730 ◽  
Author(s):  
M. Efendiev ◽  
S. Zelik ◽  
A. Miranville
Keyword(s):  
Dynamical Systems ◽  
Reaction Diffusion ◽  
Exponential Attractor ◽  
Hölder Continuous ◽  
Exponential Attractors ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional ◽  
Equations Of Mathematical Physics ◽  
External Forces ◽  
Autonomous Dynamical Systems

We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous equations of mathematical physics, the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.

scholarly journals Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains

2018 ◽  
Vol 16 (1) ◽  
pp. 862-884
Author(s):  
Xiaoyao Jia ◽  
Xiaoquan Ding ◽  
Juanjuan Gao
Keyword(s):  
Dynamical Systems ◽  
White Noise ◽  
Reaction Diffusion ◽  
Random Parameter ◽  
Reaction Diffusion Equations ◽  
Random Dynamical Systems ◽  
Diffusion Equations ◽  
Uhlenbeck Process ◽  
Random Attractors ◽  
Multiplicative White Noise

AbstractIn this paper we investigate the stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domain ℝn (n ≥ 2). We first transform the retarded reaction-diffusion equations into the deterministic reaction-diffusion equations with random parameter by Ornstein-Uhlenbeck process. Next, we show the original equations generate the random dynamical systems, and prove the existence of random attractors by conjugation relation between two random dynamical systems. In this process, we use the cut-off technique to obtain the pullback asymptotic compactness.

scholarly journals Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Xinyang Wang ◽  
Junquan Song
Keyword(s):  
Dynamical Systems ◽  
Exact Solutions ◽  
Separation Of Variables ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Financial Mathematics ◽  
Symmetry Reductions ◽  
Finite Dimensional ◽  
Wide Range

The method of conditional Lie-Bäcklund symmetry is applied to solve a class of reaction-diffusion equations ut+uxx+Qxux2+Pxu+Rx=0, which have wide range of applications in physics, engineering, chemistry, biology, and financial mathematics theory. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamical systems. The exact solutions obtained in concrete examples possess the extended forms of the separation of variables.

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